diff --git a/core/lib/Drupal/Core/DrupalKernel.php b/core/lib/Drupal/Core/DrupalKernel.php
index 26068b8..7ab72da 100644
--- a/core/lib/Drupal/Core/DrupalKernel.php
+++ b/core/lib/Drupal/Core/DrupalKernel.php
@@ -434,6 +434,7 @@ protected function preHandle(Request $request) {
    */
   public function handlePageCache(Request $request) {
     $this->boot();
+    $this->initializeCookieGlobals($request);
 
     // Check for a cache mode force from settings.php.
     if (Settings::get('page_cache_without_database')) {
@@ -886,16 +887,18 @@ protected function initializeCookieGlobals(Request $request) {
     if (static::$isRequestInitialized) {
       return;
     }
-    global $base_url, $cookie_domain;
+    global $cookie_domain;
 
     if ($cookie_domain) {
       // If the user specifies the cookie domain, also use it for session name.
       $session_name = $cookie_domain;
     }
     else {
-      // Otherwise use $base_url as session name, without the protocol
-      // to use the same session identifiers across HTTP and HTTPS.
-      list(, $session_name) = explode('://', $base_url, 2);
+      // Build an unique session name based on host and any site directory. We
+      // avoid including the protocol to use the same session identifiers across
+      // HTTP and HTTPS.
+      $session_name = $request->getHost() . $request->getBaseUrl();
+
       // HTTP_HOST can be modified by a visitor, but has been sanitized already
       // in DrupalKernel::bootEnvironment().
       if ($cookie_domain = $request->server->get('HTTP_HOST')) {